QUANTILE REGRESSION METHODS FOR RECURSIVE STRUCTURAL EQUATION MODELS

Transcription

1 QUANTILE REGRESSION METHODS FOR RECURSIVE STRUCTURAL EQUATION MODELS LINGJIE MA AND ROGER KOENKER Abstract. Tw classes f quantile regressin estimatin methds fr the recursive structural equatin mdels f Chesher (2003) are investigated. A class f weighted average derivative estimatrs based directly n the identificatin strategy f Chesher is cntrasted with a new cntrl variate estimatin methd. The latter impses strnger restrictins achieving an asympttic efficiency bund with respect t the frmer class. An applicatin f the methds t the study f the effect f class size n the perfrmance f Dutch primary schl students shws that (i.) reductins in class size are beneficial fr gd students in language and fr weaker students in mathematics, (ii) larger classes appear beneficial fr weaker language students, and (iii.) the impact f class size n bth mean and median perfrmance is negligible. 1. Intrductin Classical tw-stage least squares methds and the limited infrmatin maximum likelihd estimatr prvide attractive methds f estimatin fr Gaussian linear structural equatin mdels with additive errrs. Hwever, these methds ffer nly a cnditinal mean view f the structural relatinship, implicitly impsing quite restrictive lcatin-shift assumptins n the way that cvariates are allwed t influence the cnditinal distributins f the endgenus variables. Quantile regressin methds seek t braden this view, ffering a mre cmplete characterizatin f the stchastic relatinship amng variables and prviding mre rbust, and cnsequently mre efficient, estimates in sme nn-gaussian settings. Amemiya (1982) was the first t seriusly cnsider quantile regressin methds fr the structural equatin mdel shwing the cnsistency and asympttic nrmality f a class f tw-stage median regressin estimatrs. Subsequent wrk f Pwell (1983) and Chen and Prtny (1996) extended this apprach, but maintained the fcus primarily n the cnditinal median prblem. Recent wrk has sught t braden the perspective. Abadie, Angrist and Imbens (2002) cnsidered quantile regressin methds fr estimating endgenus treatment effects fcusing n the binary treatment case. Sakata (2000) has cnsidered a median regressin analgue f the LIML estimatr. Chernzhukv and Hansen (2001) have prpsed a nvel instrumental variables apprach. Versin: February 3, The authrs wuld like t express their appreciatin t Andrew Chesher fr stimulating cnversatins regarding this wrk. They wuld als like t thank Annelie van der Wind fr her extensive help with the interpretatin f the PRIMA data. This wrk was partially supprted by NSF Grant SES

2 2 Quantile Regressin Methds fr Structural Mdels In a series f recent papers Chesher (2001, 2002, 2003) has cnsiderably expanded the scpe f quantile regressin methds fr structural ecnmetric mdels. He cnsiders a general nnlinear specificatin whse crucial feature is its triangular stchastic structure. By recursively cnditining, a sequence f cnditinal quantile functins are available t characterize the mdel and identify the structural effects. The apprach may be viewed as a natural generalizatin f the causal chain mdels advcated by Strtz and Wld (1960). Imbens and Newey (2002) have als recently stressed the utility f the triangular stchastic structure. Chesher has elegantly laid ut the structural interpretatin f his prpsed mdels and dealt with the ensuing identificatin issues. In s ding he has clarified the bjectives f estimating mdels with hetergeneus structural effects; his fcus n structural derivatives f cnditinal quantile functins prvides a natural target fr nnparametric identificatin and estimatin. Our bjective is t cnsider mre pragmatic prblems f estimatin and inference in parametric structural mdels. We will cnsider tw general classes f the estimatin methds. The first is a class f average derivative methds based directly n the Chesher identificatin strategy. The secnd is a new cntrl variate apprach. In parametric settings we cmpare the asympttic behavir f the tw appraches and shw that the cntrl variate methds attain an efficiency bund crrespnding t an ptimally weighted frm f the average derivative estimatr. In typical applicatins where the precise specificatin f the cvariate effects are subject t dispute the tw estimatin strategies are useful cmplements, ffering a valuable framewrk fr inference. The next sectin intrduces the recursive structural mdel and describes the tw classes f estimatrs. We will fcus primarily n a simple tw equatin setting, with sme brief remarks n the extensin t larger mdels. Sectins 3 and 4 are devted t the asympttic behavir f the estimatrs and their asympttic relative efficiency. Sectin 5 reprts the results f a small simulatin experiment designed t explre the finite sample perfrmance f the tw appraches. Sectin 6 describes an applicatin f the mdels t the prblem f estimating structural effects f changes in class size n student perfrmance in Dutch primary schls. 2. Recursive Structural Mdels and Their Estimatin T mtivate Chesher s apprach it is wrthwhile t briefly recnsider the simple, exactly identified, triangular mdel, (2.1) Y i1 = Y i2 α 1 + x i α 2 + ν i1 + λν i2 (2.2) Y i2 = z i β 1 + x i β 2 + ν i2. Suppse that the unbserved errrs ν i1 and ν i2 are stchastically independent and identically distributed with ν i1 F 1 and ν i2 F 2. Assume further that the ν ij s are independent f (z i, x i ), and that fr cnvenience Y i2 and z i are scalars.

3 Lingjie Ma and Rger Kenker 3 We will fcus n the estimate f the scalar structural parameter α 1. The classical tw stage least squares estimatr f α 1 may be written as, ˆα 1 = (Ŷ 2 M XŶ2) Ŷ 2 M XY 1 where Ŷ2 = z ˆβ 1 + X ˆβ 2, ˆβ 1 = (z M X z) z M X Y 2, ˆβ 2 = (X M z X) X M z Y 2, M z = I z(z z) z, and M X = I X(X X) X. A smewhat less cnventinal interpretatin f ˆα 1 can be derived substituting fr ν i2 in (2.1) t btain (2.3) Y 1 = Y 2 (α 1 + λ) zβ 1 λ + X(α 2 λβ 2 ) + ν 1 W δ + ν 1, where W = [Y 2.z.X] and δ = (δ 1, δ 2, δ 3 ) = (α 1 + λ, β 1 λ, α 2 λβ 2 ). Nw, suppse we estimate the hybrid structural equatin (2.3) by rdinary least squares. We have the fllwing result. Prpsitin 1. ˆα 1 = ˆδ 1 + ˆβ 1 ˆδ 2, where ˆδ = (W W ) W Y 1. The prf f this result is smewhat invlved and is, therefre, relegated t the Appendix, as are prfs f subsequent results, but its interpretatin is simple and straightfrward. The tw stage least squares estimatr may be viewed as a bias crrected frm f the least squares estimatr f the structural effect in the hybrid mdel (2.3). The same strategy can be emplyed t estimate the cnditinal quantile effects in this mdel. We have the cnditinal quantile functins Q 1 (τ 1 Y 2, z, x) = Y 2 (α 1 + λ) zβ 1 λ + x (α 2 λβ 2 ) + F1 (τ 1 ) Q 2 (τ 2 x, z) = zβ 1 + x β 2 + F2 (τ 2 ). Prvided that z Q 2 (τ 2 z, x) = β 1 0 we may write fllwing Chesher (2003), α 1 = Y2 Q 1 (τ 1 Y 2, x, z) + zq 1 (τ 1 Y 2, x, z) z Q 2 (τ 2 x, z) α 2 = x Q 1 (τ 1 Y 2, x, z) zq 1 (τ 1 Y 2, x, z) x Q 2 (τ 2 x, z), z Q 2 (τ 2 x, z) adpting the cnventin that Q 1 (τ 1 Y 2, x, z) is always evaluated at Y 2 = Q 2 (τ 2 x, z). In this case, because the cvariate effects take the simple lcatin shift frm, the structural parameters α 1 and α 2 are glbally cnstant independent f τ 1 and τ 2 and f the exgenus variables x and z. As we will nw see, this is highly unusual Quantile Treatment Effects fr Recursive Structural Mdels. Nw cnsider the nnlinear recursive mdel (2.4) Y i1 = ϕ 1 (Y i2, x i, ν i1, ν i2 ) (2.5) Y i2 = ϕ 2 (z i, x i, ν i2 ) where as earlier we assume that ν i1 and ν i2 are independent, and identically distributed with ν ij F j. The pairs (ν i1, ν i2 ) are als maintained t be independent f (z i, x i ). The functin ϕ 1, is assumed strictly mntnic in ν 1, and differentiable with

4 4 Quantile Regressin Methds fr Structural Mdels respect t Y 2 and x, and ϕ 2 is assumed strictly mntnic in ν 2, and differentiable with respect t bth z and x. Under these cnditins, we can write the cnditinal quantile functins, Q 1 (τ 1 Q 2 (τ 2 x, z), x) = ϕ 1 (Q 2 (τ 2 x, z), x, F1 (τ 1 ), F2 (τ 2 )) Q 2 (τ 2 x, z) = ϕ 2 (z, x, F2 (τ 2 )). Hw shuld we measure the effect f Y 2 n Y 1 in this mdel? Given the stchastic character f the treatment, Y 2, we must evaluate the treatment effect at varius quantiles f the treatment distributin. We may view this as crrespnding t a thught experiment in which we exgenusly alter nt the value f Y 2 as we wuld with a treatment fully under ur cntrl, but instead alter the distributin f Y 2. Thus, fr example, in ur anticipated study f class-size effects n educatinal perfrmance, we may imagine altering the prevailing distributin f class-sizes and explring the cnsequences f this perturbatin n varius quantiles f the distributin f students attainment. Of curse, in the mdel Y 2 is determined accrding t (2.5), s t assume therwise requires sme srt f willing suspensin f disbelief in the mdel. But this is inevitable in structural mdels and we are always entitled t interpret effects as lng as they can be frmulated in terms f well-psed gedankan experiments. In their (infamus) tryptych n causal chain systems Strtz and Wld (1960) illustrate this pint with a vivid fresh water example: Suppse z is a vectr whse varius elements are the amunts f varius fish feeds (different insects, weeds, etc.) available in a given lake. The reduced frm y = B Γz + B u wuld tell us specifically hw the number f fish f any species depends upn the availabilities f different feeds. The cefficient f any z is the partial derivative f a species ppulatin with respect t a fd supply. It is t be nted, hwever, that the reduced frm tells us nthing abut the interactins amng the varius fish ppulatins it des nt tell us the extent t which ne species f fish feeds n anther species. Thse are the causal relatins amng the y s. Suppse, in anther situatin, we cntinuusly restck the lake with species g, increasing y g by any desired amunt. Hw will this affect the values f the ther y s? If the system were recursive and we had estimates f the elements f B, we wuld simply strike the gth equatin ut f the mdel and regard y g, the number f fish f species g, as exgenus as a fd supply r, when appearing with a negative cefficient as a pisn. (pp , emphasis added) Recursive cnditining enables us t cntemplate similar kinds f plicy experiments in the cntext f the triangular structural mdels cnsidered by Chesher; related mdels have als been recently cnsidered by Imbens and Newey (2002). In cntrast t the linear structural mdels f the Cwles Cmmissin era, whse causal

5 Lingjie Ma and Rger Kenker 5 effects were restricted t take the frm f lcatin shifts f the cnditinal distributins f the endgenus variables, recent wrk pses the identificatin f structural effects in a general nn-parametric framewrk s structural effects can take quite hetergeneus frms. We will fcus n a mre restricted finite dimensinal parametric frmulatin, a frmulatin that is mre cnducive t ur asympttic analysis. Extensins t sequences f mdels with the parametric dimensin tending t infinity culd be cnsidered in subsequent wrk. T explre this further, cnsider the fllwing mdel in which Y 2 exerts bth a lcatin and a scale shift effect n Y 1 ; (2.6) Y i1 = Y i2 α 1 + x i α 2 + δy i2 (ν i1 + λν i2 ) (2.7) Y i2 = z i β 1 + x i β 2 + γz i ν i2. Maintaining ur prir assumptins n (ν i1, ν i2 ), and assuming that δ 0 and γ 0, we can again substitute fr ν i2 in (2.6) t btain, Y i1 = Y i2 (α 1 + δν i1 δβ 1 λ/γ) + x i α 2 + Y ( ) i2 2 δλ Y ( ) i2x i δλβ2, z i γ z i γ Y i2 = z i (β 1 + γν i2 ) + x i β 2. S we have the cnditinal quantile functins, (2.8) (2.9) Q 1 (τ 1 Y i2, x i, z i ) = Y i2 θ 1 (τ 1 ) + x i θ 2(τ 1 ) + Y 2 i2 z i θ 3 (τ 1 ) + Y i2x i z i θ 4 (τ 1 ) Q 2 (τ 2 x, z) = zβ(τ 2 ) + x β 2 (τ 2 ), where θ 1 (τ 1 ) = α 1 +δf1 (τ 1 ) δβ 1 λ/γ, θ 2 (τ 1 ) = α 2, θ 3 (τ 1 ) = δλ/γ, θ 4 (τ 1 ) = δλβ 2 /γ, β 1 (τ 2 ) = β 1 + γf2 (τ 2 ) and β 1 (τ 2 ) = β 2. By recursive cnditining we have the cnditinal quantile functins, Q 1 (τ 1 Q 2 (τ 2 x, z), x, z) = Q 2 (τ 2 x, z)(α 1 + δ(f1 (τ 1 ) + λf2 (τ 2 ))) + x α 2 Q 2 (τ 2 x, z) = z(β 1 + γf2 (τ 2 )) + x β 2 s the structural effect f interest is, A straightfrward calculatin shws that π 1 (τ 1, τ 2 ) = α 1 + δ(f 1 (τ 1 ) + λf 2 (τ 2 )). π 1 (τ 1, τ 2 ) = Y2 Q 1 (τ 1 Y 2, x, z) + zq 1 (τ 1 Y 2, x, z). z Q 2 (τ 2 x, z) As in the lcatin-shift mdel, this structural effect is independent f the cnditining cvariates x i and z i s the Chesher identificatin strategy suggests an bvius estimatin strategy. Nte hwever, that since estimatin f the cnditinal quantile functins (2.8) and (2.9) will fail t prduce the cnvenient cancellatin f the exact calculatin, sme scheme t average ver the cvariate space wuld be required t btain the structural effect. This will be even mre apparent in the next subsectin where a mre general nnlinear-in-parameters mdel is cnsidered.

6 6 Quantile Regressin Methds fr Structural Mdels Given the separate cntributins f F1 (τ 1 ) and F2 (τ 2 ), it is clear that π(τ 1, τ 2 ) reflects nt nly the fact that the stchastic effect f Y 2 n Y 1 arises frm tw distinct surces, but als prvides structural insight int hw these surces are related. Suppse we fix τ 1 s ν 1 is fixed at its τ 1 quantile, changes in τ 2 in π 1 (τ 1, τ 2 ) reflect hw the distributin f ν 2 affects the τ 1 quantile f the respnse Y 1. On the ther hand, if we fix τ 2, and allw τ 1 t change, this sheds light n hw the τ 2 quantile f Y 2 influences the whle distributin f the respnse Y 1. By cnsidering variatin in bth τ 1 and τ 2 we btain a panramic view f the stchastic relatinship between Y 2 and Y 1. Recalling that integrating the quantile functin F X (τ) f a randm variable, X, ver the dmain [0, 1], yields its expectatin, that is, EX = 1 0 F (t)dt, X we can define a mean quantile treatment effect by integrating ut τ 2, and denting µ i = Eν i, π 1 (τ 1 ) = 1 0 (α 1 + δ(f 1 (τ 1 ) + λf 2 (τ 2 )))dτ 2 α 1 + δf 1 (τ 1 ) + δλµ 2 Averaging again, this time with respect t τ 1 yields the mean treatment effect π 1 = 1 0 (α 1 + δf 1 (τ 1 ) + δλµ 2 )dτ 1 α 1 + δµ 1 + δλµ 2. This mean treatment effect wuld be what is estimated by the tw stage least squares estimatr in the pure lcatin shift versin f the mdel, but when the effects are mre hetergeneus as in this lcatin-scale shift mdel the structural quantile treatment effect π 1 (τ 1, τ 2 ) represents a decnstructin the mean effect int its elementary cmpnents. Figure 2.1 illustrates the three versins f the treatment effect π 1 (τ 1, τ 2 ), π 1 (τ 1 ) and π 1 fr a particular parametric instance f the mdel (2.6-7) Estimatin f Structural Quantile Treatment Effects. In this sectin we will describe tw general classes f estimatrs fr the parametric recursive structural mdel, (2.10) (2.11) Y i1 = ϕ 1 (Y i2, x i, ν i1, ν i2 ; α) Y i2 = ϕ 2 (z i, x i, ν i2 ; β). We will maintain ur assumptins n the ν ij s and the functins ϕ 1 ϕ 2 and we will explicitly assume that the functins ϕ 1 and ϕ 2 are knwn up t the finite dimensinal parameter vectrs α and β. Under these cnditins we have an inverse functin fr ϕ 2 with respect t ν 2, say ϕ 2, allwing us t write and thus we have, ν i2 = ϕ 2 (Y i2, z i, x i ; β) Y i1 = ϕ 1 (Y i2, x i, ν i1, ϕ 2 (Y i2, z i, x i ; β); α).

7 a a1(1) a1(1, 2) Lingjie Ma and Rger Kenker 7 Mean Treatment Effect Mean Quantile Treatment Effect Quantile Treatment Effect Figure 2.1. Quantile Treatment Effects fr the Structural Mdel: The figure illustrate three different ntins f the structural treatment effect fr the linear lcatin-scale structural equatin mdel: (2.6-7) with (α 1, α 2, δ, λ) = (10, 4, 3, 2), (β 1, β 2, γ) = (1, 2, 3), ν 1 N(0, 1), ν 2 N(0, 0.5). The left figure depicts π 1 =10, the mean treatment effect; the middle figure shws π 1 (τ 1 ) = 10+3F1 (τ 1 ), the mean quantile treatment effect; the right figure shws π 1 (τ 1, τ 2 ) = (F1 (τ 1 ) + 2F2 (τ 2 )), the general quantile treatment effect. We will write the cnditinal quantile functins f Y 1 and Y 2 as, Q 1 (τ 1 Y i2, x i, z i ) = h 1 (Y i2, x i, z i ; θ) Q 2 (τ 2 z i, x i ) = h 2 (z i, x i ; β). Fixing τ 1 and τ 2 we can estimate the parameters f the cnditinal quantile functins, θ(τ 1 ) and β(τ 2 ), as illustrated in the previus subsectin, by slving the pssibly nnlinear weighted quantile regressin prblems, (2.12) (2.13) ˆθ(τ 1 ) = argmin θ Θ ˆβ(τ 2 ) = argmin β B n σ i1 ρ τ1 (Y i1 h 1 (Y i2, x i, z i ; θ)) n σ i2 ρ τ2 (Y i2 h 2 (z i, x i, β)). The weights σ ij are assumed t be strictly psitive and will play an imprtant rle in the efficiency cmparisns made in Sectin 4. The functin ρ τ (u) = u(τ I(u < 0)) is as in Kenker and Bassett (1978). Methds fr cmputing quantile regressin estimates fr mdels that are nnlinear in parameters are described in Kenker and Park (1996). When h 1 and h 2 yield specificatins that are nnlinear in parameters, then we require cmpact dmains Θ and B fr the parameters. Our primary bjective will be t estimate the weighted average quantile treatment effect implied by the Chesher frmula, { π 1 (τ 1, τ 2 ) = y Q 1 (τ 1 y, x i, z i ) + } zq 1 (τ 1 y, x i, z i ) w(x, z)dxdz z Q 2 (τ 2 x i, z i )

8 8 Quantile Regressin Methds fr Structural Mdels with y evaluated as befre, at Q 2 (τ 2 x i, z i ). A secndary bject will be t estimate the crrespnding structural effect f the exgnus variables x, { π 2 (τ 1, τ 2 ) = x Q 1 (τ 1 Y i2, x i, z i ) } zq 1 (τ 1 y, x i, z i ) z Q 2 (τ 2 x i, z i ) xq 2 (τ 2 x i, z i ) w(x, z)dxdz Since, in general, the abve integrands depend upn the pint f evaluatin in the space f the exgenus cvariates we cnsider the class f weighted average derivative estimatrs, { n ˆπ 1 (τ 1, τ 2 ) = w i y ĥ 1 (τ 1 y, x i, z i, ˆθ) + zĥ1(τ 1 y, x i, z i, ˆθ) } z ĥ 2 (τ 2 x i, z i. ˆβ) again evaluating at y = ĥ2(τ 2 x i, z i, ˆβ). A weighted average derivative estimatr fr the structural effects f x is defined similarly as, { n ˆπ 2 (τ 1, τ 2 ) = w i x ĥ 1 (τ 1 y, x i, z i, ˆθ) zĥ1(τ 1 y, x i, z i, ˆθ) } z ĥ 2 (τ 2 x i, z i, ˆβ) xĥ2(τ 2 x i, z i, ˆβ). The weights are assumed t be psitive and sum t ne. A cnvenient chice wuld be w i n. In sme cases, like the lcatin shift mdel the dependence n the exgenus cvariates vanishes s the weights are irrelevant. The freging cnsideratins have presumed a situatin f exact identificatin in which there is a single instrumental variable, z, available. In ver-identified settings we may have several versins f ˆπ(τ 1, τ 2 ) crrespnding t different chices f the variable z and we may wish t again cnsider weighted averages. This pint will be addressed in mre detail when we cme t asympttics. The estimatr ˆπ n (τ 1, τ 2 ) = (ˆπ 1 (τ 1, τ 2 ), ˆπ 2 (τ 1, τ 2 )) is based squarely n Chesher s identificatin strategy. Its advantage is that it takes a rather skeptical attitude tward the riginal mdel and is thereby based n a rather lsely restricted frm f the tw cnditinal quantile functins. This cmplements nicely the mre restrictive frm f the estimatrs described in the next subsectin and cnsequently may eventually prve t be advantageus frm a specificatin diagnstics and testing viewpint A Cntrl Variate Estimatr. T mtivate the cntrl variate apprach t estimatin f the structural quantile treatment effect, it is helpful t return briefly t the classical tw stage least squares estimatr f the lcatin shift mdel (2.1-2) and recall its cntrl variate interpretatin. Suppse that rather than replacing Y 2 by Ŷ 2 in (2.1) and estimating the resulting mdel by least squares, we instead cmpute ˆν 2 = Y 2 Ŷ2, the residuals frm the first stage f 2SLS. Nw cnsider including ˆν 2 as an additinal cvariate in (2.1) and estimating by least squares. It is easy t shw that the resulting estimates f α 1 and α 2 are the same as thse prduced by 2SLS. This result hlds much mre generally: Y i2 and z i may be vectr-valued and the mdel may be veridentified. A definitive riginal reference fr this equivalence is hwever difficult t identify, see fr example, Blundell and Pwell (2003).

9 Lingjie Ma and Rger Kenker 9 T apply the cntrl variate apprach t the estimatin f the structural quantile treatment effect we must first estimate the cnditinal τ 2 quantile functin f Y 2 t recver an estimate f ν 2 (τ 2 ) = ν 2 F2 (τ 2 ). Let Q 1 (τ 1 Y i2, x i, ν i2 (τ 2 )) = g 1 (Y i2, x i, ν i2 (τ 2 ); α(τ 1, τ 2 )) Q 2 (τ 2 z i, x i ) = g 2 (z i, x i ; β(τ 2 )) dente the cnditinal quantile functins f the respnse variables cnditining n the cntrl variate, ν i2 (τ 2 ). Slving ˆβ(τ 2 ) = argmin σi2 ρ τ2 (Y i2 g 2 (z i, x i ; β)) β B ur cnditins n ϕ 2 insure that we can invert t btain the functin ν 2 = ϕ 2 (Y 2, z, x, β) s and we have F 2 (τ 2 ) = ϕ 2 (g 2 (z, x; β), z, x; β) ˆν i2 (τ 2 ) = ϕ 2 (Y i2, z i, x i ; ˆβ) ϕ 2 (g 2 (z i, x i ; ˆβ), z i, x i ; ˆβ). Nte that the abve prcedure is valid regardless f the dimensin f z i, s as lng as the mdel is identifiable ˆν i2 (τ 2 ) incrprates infrmatin n all f the available instruments. But it des s in a much mre parsimnius fashin than by intrducing z i directly int what we have referred t as the hybrid frm f the first structural equatin. Once ˆν i2 (τ 2 ) is available we can estimate the parameters f the first structural equatin by reexpressing ϕ 1 as g 1 (Y i2, x i, ˆν i2 (τ 2 ); a) = ϕ 1 (Y i2, x i, F 1 (τ 1 ), ˆν i2 (τ 2 ); α) absrbing F 1 (τ 1 ) int the new parameter vectr a, and slving, ˆα(τ 1, τ 2 ) = argmin a A n σ i1 ρ τ1 (Y i1 g 1 (Y i2, x i, ˆν i2 (τ 2 ); a)). In the next sectin we will investigate the asympttic behavir f this cntrl variate estimatr and cmpare its asympttic perfrmance with the weighted average derivative estimatr. Befre ding s we might remark that the restrictins impsed by the cntrl variate prcedure avid the cnsiderable cmplicatins f the weighted average derivative methd apparent in the lcatin-scale mdel (2.6-7).

10 10 Quantile Regressin Methds fr Structural Mdels 2.4. Extensin t m Equatins. As shwn by Chesher (2003) there are n real impediments t the extensin f the recursive structural mdel t mre than tw equatins, except sme bvius ntatinal nes. Maintaining the triangular structure we may cnsider the system f m structural equatins, Y 1 = ϕ 1 (Y 2,..., Y m, z, ν 1,..., ν m, α 1 ) Y 2 = ϕ 2 (Y 3,..., Y m, z, ν 2,..., ν m, α 2 ). Y m = ϕ m (z, ν m, α m ). The ν s are assumed stchastically independent and independent f the exgnus variables, z. Again, we can recursively cnditin t btain the cnditinal quantile functins f the Y s and this leads t a natural generalizatin f the weighted average derivative estimatrs. Chesher (2003) describes the exclusin restrictins and ther cnditins required fr identificatin in this case. Similarly, we can adapt the cntrl variate estimatin methd t the multiple equatin setting. The estimatin strategy is a quite straightfrward extensin f the tw equatin situatin. Starting with the last equatin we estimate the cntrl variate ˆν m (τ m ) and substitute it int the (m)th equatin, thus btaining the cntrl variate ˆν m (τ m ), and s frth. The asympttic representatin als generalizes in a straightfrward fashin s that fr the first equatin, fr example, we btain a sum f m independent terms in the Bahadur representatin. 3. Asymptpia The asympttic behavir f the estimatrs described in the previus sectin can be develped with the aid f existing results n the asympttics f nnlinear (in parameters) quantile regressin estimatin. We will maintain the cnditins set ut fllwing the general mdel specificatin (2.4) and (2.5) and its parametric frmulatin (2.10) and (2.11). In additin we will emply the fllwing regularity cnditins: as, e.g., in Oberhfer (1982) and Jurečkvá and Prcházka(1994). A.1: The cnditinal distributin functins F Y1 (y 1 Y i2, x i, z i ) and F Y2 (y 2 z i, x i ) are abslutely cntinuus with cntinuus densities f i1 and f i2 that are unifrmly bunded away frm 0 and at the pints ξ i1 = Q 1 (τ 1 Q 2 (τ 2 z i, x i ), x i, z i ) and ξ i2 = Q 2 (τ 2 z i, x i ), fr i = 1,..., n. The weights σ ij are psitive and unifrmly bunded away frm 0 and. A.2: There exist psitive definite matrices J 1, J 1, J 2, J 2 such that lim n n σ 2 ijḣijḣ ij = J j, lim n n σ ij f ij (ξ ij )ḣijḣ ij = J j, where ḣi1 = θ h i1 and ḣi2 = β h i2. A.3: max,...,n ḣij / n 0, j = 1, 2.

11 Lingjie Ma and Rger Kenker 11 A.4: There exist cnstants l 1, l 2, u 1, u 2 and an integer n 0 > 0 such that fr (θ j, θ j ) Θ, (β j, β j ) B, j = 1, 2 and n > n 0, l 1 θ θ (n (h 1 (Y i2, x i, z i, θ) h 1 (Y i2, x i, z i, θ ) 2 ) 1/2 u 1 θ θ l 2 β β (n (h 2 (x i, z i, β) h 2 (x i, z i, β ) 2 ) 1/2 u 2 β β. Therem 1. Fr the parametric mdel ( ) satisfying cnditins A.1-4, the weighted average derivative estimatr ˆπ n (τ 1, τ 2 ) has the asympttic linear (Bahadur) representatin n(ˆπn (τ 1, τ 2 ) π(τ 1, τ 2 )) = W 1 J 1 n/2 + W 2 J 2 n/2 n σ i1 ḣ i1 ψ τ1 (Y i1 ξ i1 ) n σ i2 ḣ i2 ψ τ2 (Y i2 ξ i2 ) + p (1) N (0, ω 11 W 1 J 1 J 1 J 1 W 1 + ω 22 W 2 J 2 J 2 J 2 W 2 ) where ω jj = τ j (1 τ j ), W 1 = θ π(τ 1, τ 2 ) and W 2 = β π(τ 1, τ 2 ). Remark: It is immediately apparent that the ptimal chice f the weights, σ ij invlves setting σ ij = f ij (ξ ij ). In this case the sandwich frm f the limiting cvariance matrix simplifies, and we have n(ˆπn (τ 1, τ 2 ) π(τ 1, τ 2 )) N (0, ω 11 W 1 J1 W 1 + ω 22W 2 J2 W 2 ). Newey and Pwell (1990) have shwn that this density weighting achieves a semiparametric efficiency bund fr a class f linear quantile regressin mdels. We will nt address the smewhat delicate issues invlved in estimating weights, but the interested reader culd cnsult Kenker and Zha (1994) and/r Zha (2001). Example: Recall that in the pure lcatin shift versin f the mdel (2.1-2) the structural effect π 1 (τ 1, τ 2 ) is a cnstant α 1. In this case we have mdel (2.1-2) and n(ˆα1 (τ 1, τ 2 ) α 1 ) is asympttically Gaussian with mean 0 and variance ( τ1 (1 τ 1 ) v = f1 2 (ξ 1 ) + λ 2 τ ) 2(1 τ 2 ) v f2 2 0 (ξ 2 ) where v 0 = lim n n β 1Z M X Zβ 1, and M X = I X(X X) X. The parameter λ may be interpreted as a degree f endgeneity f the mdel, s the secnd term in v may be viewed as a perfrmance penalty fr this endgeneity effect. It may be nted that under these special cnditins the estimatr ˆα 1 (τ 1, τ 2 ) is equivalent t the s-called tw-stage quantile regressin estimatr which replaces Y 2 in (2.1) by Ŷ2(τ 2 ) the fitted values in the τ 2 quantile regressin estimate f (2.2) and then estimates the τ 1 quantile regressin f Y 1 n Ŷ2(τ 2 ) and x. A special case f this prcedure is Amemiya s tw stage least abslute deviatins estimatr. T the best f ur knwledge n general analysis f its asympttic behavir has been undertaken althugh it has been emplyed in several empirical studies.

12 12 Quantile Regressin Methds fr Structural Mdels T study the asympttic behavir f the cntrl variate estimatrs we require a slightly mdified versin f ur previus regularity cnditins. B.1: The cnditinal distributin functins F Yi1 Y i2,x i,ν i2 and F Yi2 z i,x i are abslutely cntinuus with cntinuus densities f i1 and f i2 unifrmly bunded away frm 0 and at the pints ξ i1 = Q 1 (τ 1 Y i2, z i, x i ), x i, ν(τ 2 )) and ξ i2 = Q 2 (τ 2 z i, x i ), respectively fr i = 1, 2,..., n. The weights σ ij are psitive and unifrmly bunded away frm 0 and. B.2: There exist psitive definite matrices D 1, D 1, D 2, D 2 such that lim n n σijġ 2 ij ġij = D j, lim n σ ij f ij (ξ ij )ġ ij ġij = D j, n where ġ i1 = α g i1 and ġ i2 = β g i2. B.3: max,...,n ġ ij / n 0, j = 1, 2. B.4: There exist cnstants l 1, l 2, u 1, u 2 and an integer n 0 > 0 such that such that, fr α, α A, β, β B and n > n 0, n l 1 α α (n (g 1 (Y i2, x i, ν i2 (τ 2 ), α) g 1 (Y i2, x i, ν i2 (τ 2 ), α )) 2 ) 1/2 u 1 α α l 2 β β (n n (g 2 (x i, z i, β) g 2 (x i, z i, β)) 2 ) 1/2 u 2 β β. These cnditins are the natural analgues f ur previus cnditins. It may be nted that in cntrast t the prir cnditins, hwever, the pssibility f veridentificatin is nw permitted by the mdified cnditins. We can nw describe the asympttic behavir f the cntrl variate estimatr. Therem 2. Fr the parametric mdel ( ) satisfying cnditins B.1-4, the cntrl variate estimatr ˆα n (τ 1, τ 2 ) has the asympttic linear (Bahadur) representatin, n n(ˆαn (τ 1, τ 2 ) α(τ 1, τ 2 )) = D 1 n /2 σ i1 ġ i1 ψ τ1 (Y i1 ξ i1 ) + D 1 D 12 D 2 n/2 N (0, ω 11 D 1 D 1 n σ i2 ġ i2 ψ τ2 (Y i2 ξ i2 ) + p (1) D 1 + ω D 22 1 D D 12 2 D 2 D 2 D 12 D 1 ) where D 12 = lim n n σ i1 f i1 η i ġ i1 ġ i2 and η i = ( g 1i / ν i2 (τ 2 ))( νi2 ϕ i2 ). Remark: Again, we see that the chice f the weights σ ij = f ij (ξ ij ) is ptimal. It may appear that the use f symbls σ ij fr the weights fr bth classes f estimatrs is an abuse f ntatin, but careful examinatin f the cnditining reveals that the cnditinal densities are identical in cnditins A.1 and B.1 s this ecnmy is justified at least in the tw cases f primary interest: weights identically equal t ne, and ptimally weighted estimatin accrding t the cnditinal densities. Fr purpses f inference it is crucial that we have nt nly the marginal distributin f ˆα n fr fixed τ 1 and τ 2, but als the jint distributin f ˆα n evaluated at several

13 Lingjie Ma and Rger Kenker 13 τ 1 s and τ 2 s. But this fllws immediately frm the Bahadur representatin f the preceding therem. Crllary 1. Let T 1 = {τ 11,...τ 1q } and T 2 = {τ 21,...τ 2r } with elements τ ij (0, 1), then under the cnditins f Therem 2, the jint asympttic distributin f {ˆα n (τ 1, τ 2 ) : τ 1 T 1, τ 2 T 2 } is Gaussian with typical cvariance blck, Acv ( nˆα(τ 1, τ 2 ), nˆα(τ 3, τ 4 ) ) = ω D 13 1 D 13 D 3 + ω D 24 1 D D 12 2 D 24 D 4 D 34 D 3, where D rs = lim n n n σ irσ is ġ ir ġ is, ω rs = min{τ r, τ s } τ r τ s, with {τ 1, τ 3 } T 1 and {τ 2, τ 4 } T Asympttic Relative Efficiency f the Structural Estimatrs Naturally, we wuld like t cmpare the perfrmance f ur tw classes f estimatrs. The first and mst bvius prerequisite fr this is t ensure that they are really estimating the same quantity. Fr linear in parameters specificatins the situatin is quite straightfrward s we will cnsider this case in sme detail first, treating it as a rehearsal fr the general result embdied in Therem 4. T frmalize what we mean by linear mdels, suppse that (4.1) (4.2) ϕ 1 (Y i2, x i, ν i2, α, F1 (τ 1 )) = ġi1 α(τ 1, τ 2 ) = ḣ i1 θ(τ 1) ϕ 2 (z i, x i, F2 (τ 2 ), β) = ġi2β(τ 2 ) = ḣ i2β(τ 2 ) where the vectrs ġ ij and ḣij are free f dependence n the parameters. The linearity f ϕ 1 implies that there is a linear mapping, W 1 = π/ θ, such that W 1 θ = π. Writing G j fr the matrix with typical rw n /2 (f ij ġ ij) fr j = 1, 2, and similarly let H j dente the matrix with typical rw n /2 (f ij ḣ ij). Nte that G 2 = H 2 and that there is a matrix A such that G 1 = H 1 A s Aα = θ. Thus we have W 1 Aα = π. Further, let L = W 1 A, s Lα = π. The transfrmatin L reduces the dimensinality f the α vectr, eliminating the cmpnents that are required t describe the ν 2 -effect and allwing us t fcus attentin n the perfrmance f the cntrl variate estimatr f the π parameter. We can nw cmpare the perfrmance f ur tw estimatrs f π: the weighted average derivative estimatr ˆπ n and the cntrl variate estimatr π n = Lˆα n. T facilitate this cmparisn it is cnvenient t restrict attentin t the ptimally weighted frm f bth estimatrs fr which σ ij = f ij. In this case, the asympttic cvariance matrix f ˆπ n specializes t while that f ˆα n specializes t Avar( nˆπ n ) = ω 11 W 1 J 1 W 1 + ω 22 W 2 J 2 W 2 Avar( nˆα n ) = ω 11 D1 + ω 22 D1 D 12D2 D 12D1

14 14 Quantile Regressin Methds fr Structural Mdels where D i = lim n n f 2 ijġ ij ġ ij and D 12 = lim n n f 2 i1η i ġ i1 ġ i2. Equivalently, we can write, Avar( nˆα n ) = ω 11 (G 1 G 1) + ω 22 (G 1 G 1) G 1 P G 2G 1 (G 1 G 1) where P G generically dentes the prjectin G(G G) G nt the clumn space f the matrix G. Thus, π = Lˆα, we have, Nte that Avar( n π) = ω 11 L(G 1 G 1) L + ω 22 L(G 1 G 1) G 1 P G 2G 1 (G 1 G 1) L L(G 1 G 1 ) L = W 1 A(A H1 H 1 A) A W1 = W 1 J1 1 H 1A(A H1 H 1A) AH1 H 1J1 1 = W 1 J1 1 P G 1 H 1 J1 1 W 1 J1 1, where signifies the cnventinal rdering f matrices in the sense f psitive definite differences. Similarly, we have, L(G 1 G 1) G 1 P G 2G 1 (G 1 G 1) L W 2 J 2 W 2, s we have established that the cntrl variate estimatr, π n, has smaller asympttic variance than the weighted average derivative estimatr ˆπ n. The efficiency advantage f the cntrl variate estimatr clearly derives frm the mre restricted frm f the estimatr. While the restricted frm f the π n estimatr yields an efficiency gain when we are cnfident abut the mdel specificatin, it clearly ffers sme disadvantages in situatins in which we are nt s cnfident. Indeed, tests f mdel specificatin based n the unrestricted frm f the estimatrs (ˆθ n, ˆβ n ) might be viewed as a reasnable precautin in the early stages f mdel cnstructin. When the mdel is nnlinear in parameters the situatin is much the same frm an asympttic viewpint. Jacbians f the nnlinear transfrmatins, W 1, A, and L evaluated at the true parameters nw play the rle f the matrices in the previus develpment, and the δ-methd yields the fllwing general result. Therem 3. Fr the parametric mdel (2.4-5) with the ptimal weighting, σ ij = f ij, let Λ(α) = π dente the mapping frm the structural parameter α t the weighted average derivative parameter π. Suppse that the Jacbian, L = Λ/ α is cntinuus in a neighbrhd f the true parameters. Then the ptimally-weighted average derivative estimatr, ˆπ n, and the ptimally-weighted cntrl variate estimatr, π n = Λ(ˆα n ), have limiting Gaussian behavir with asympttic cvariance matrices: Avar( nˆπ n ) = ω 11 W 1 J 1 W 1 + ω 22W 2 J 2 W 2 Avar( n π) = ω 11 L(G 1 G 1) L + ω 22 L(G 1 G 1) G 1 P G 2G 1 (G 1 G 1) L and Avar( n π n ) Avar( nˆπ). Remark: It is wrth emphasizing at this pint that the superir asympttic perfrmance f the cntrl variate estimatr asserted in Therem 3 is particularly appealing

15 Lingjie Ma and Rger Kenker 15 when the mdel is veridentified. In such cases the weighted average derivative apprach becmes smewhat cumbersme, while the cntrl variate methd remains entirely straightfrward. 5. Mnte-Carl In this sectin we very briefly reprt n sme simulatin experiments designed t evaluate the perfrmance f the estimatin methds cnsidered abve. The cmputatinal results reprted in this and the fllwing sectin were carried ut in the R language, Ihaka and Gentleman (1996) using the quantile regressin package f Kenker (1998). We cnsider a simple lcatin-scale shift mdel: (5.1) (5.2) Y 1 = α 1 + α 2 x + (α 3 + δ(λν 2 + ν 1 ))Y 2 Y 2 = β 1 + β 2 x + β 3 z + ν 2 where x, z, ν 1 and ν 2 are generated as the fllwing: x t 3, z N(15, 2 2 ), ν 1 N(0, 1). and ν 2 N(0, ), We specify the parameter vectrs as fllwing, (α 1, α 2, α 3, δ, λ) = (3, 4, 4, 5, 3), and (β 1, β 2, β 3 ) = (1, 2, 3). Fr this mdel, bth the weighted average derivative (WAD) and the cntrl variate (CV) estimatrs fr the structural quantile treatment effect f Y 2 n Y 1 will cnverge t the ppulatin value f Fν 2 (τ 2 ) + 5Fν 1 (τ 1 ). Fr the sake f simplicity, we set τ 1 = τ 2 = τ and cnsider nly the quantiles τ = (0.1, 0.3, 0.5, 0.7, 0.9). Results are reprted in Table 5.1 fr sample size n = 100, and in Table 5.2 fr n = The number f replicatins is R = We see first, that bth estimatrs exhibit very mdest bias at sample size, n = 100, and bias is substantially reduced at n = Secndly, in terms f the standard errr and rt mean square errr, the cntrl variate estimatr utperfrms the weighted derivative estimatr at all cnsidered quantiles. Fr the sake f cmparisn we cnsider fur ther estimatrs: QR: Naive quantile regressin applied t (5.1) withut any attempt t deal with the endgneity f Y 2. 2SQRQ: Tw stage quantile regressin replacing Y 2 by the predicted Ŷ2 frm the τ = τ 2 quantile regressin estimatin f (5.2). 2SQRA: Tw stage quantile regressin replacing Y 2 by the predicted Ŷ2 frm the τ = 1/2 median regressin estimatin f (5.2). 2SQRS: Tw stage quantile regressin replacing Y 2 by the predicted Ŷ2 frm the rdinary least squares (mean) regressin estimatin f (5.2). The perfrmance f the ther estimatrs is quite unsatisfactry by cmparisn with the WADQR and CVQR prpsals. At the median the tw-stage methds all have gd bias and variance perfrmance, as ne wuld expect frm the results f Amemiya (1982). But at all ther quantiles they exhibit serius bias prblems. Bias f the varius 2SQR estimatrs is nt substantially imprved by the increase in sample size, cntrary t the perfrmance f the CVQR and WADQR estimatr. Naive quantile regressin estimatin f the structural equatin, as expected, is als

16 16 Quantile Regressin Methds fr Structural Mdels Ceffcient Bias Std. Errr RMSE τ 1 = τ 2 = 0.1 True Value CVQR WADQR SQRQ SQRA SQRS QR τ 1 = τ 2 = 0.3 True Value CVQR WADQR SQRQ SQRA SQRS QR τ 1 = τ 2 = 0.5 True Value CVQR WADQR SQRQ SQRA SQRS QR τ 1 = τ 2 = 0.7 True Value CVQR WADQR SQRQ SQRA SQRS QR τ 1 = τ 2 = 0.9 True Value CVQR WADQR SQRQ SQRA SQRS QR Table 5.1. Simulatin Results: n = 100, R = badly biased, except (ddly) at τ = 0.9, where cuntervailing bias effects seem t frtuitusly cancel.

17 Lingjie Ma and Rger Kenker 17 Ceffcient Bias Std. Errr MSE t = 0.1 True Value CVQR WADQR SQRQ SQRA SQRS QR t = 0.3 True Value CVQR WADQR SQRQ SQRA SQRS QR t = 0.5 True Value CVQR WADQR SQRQ SQRA SQRS QR t = 0.7 True Value CVQR WADQR SQRQ SQRA SQRS QR t = 0.9 True Value CVQR WADQR SQRQ SQRA SQRS QR Table 5.2. Simulatin Results: n = 1000,, R = The Effect f Class Size n Student Perfrmance in Dutch Primary Schls In this sectin we recnsider an applicatin f Levin (2001) investigating the effect f class size n student perfrmance in Dutch primary schls. We will apply bth

18 18 Quantile Regressin Methds fr Structural Mdels weighted derivative and the cntrl variate methds t a structural equatin mdel f the impact f class size n student achievement. Our main bjective is t demnstrate hw these new appraches can be emplyed t reveal new aspects f the sample and thus yield mre detailed and cnstructive plicy analysis. We find that the tw methds prduce quite similar results, especially fr language perfrmance, a finding that smewhat reenfrces ur cnfidence in ur mdel specificatin. Bth estimatrs indicate that the class size effects vary significantly acrss quantiles f the class size distributin and student achievement distributin. Fr the lwer attainment students, bigger classes imprve language perfrmance, while smaller classes imprve math scres. Fr average students, class sizes have insignificant effects n bth language and math perfrmance. Fr high attainment students smaller classes are slightly better fr language perfrmance, but class size effects are nt significant fr math perfrmance. These findings suggest that a general plicy f class size reductin is unlikely t have large beneficial effects n verall student achievement and shuld be apprached with sme skepticism A Brief Review f the Literature n Class Size Effect. Student academic perfrmance is f paramunt imprtance t parents, teachers and educatinal plicy makers. Amng plicy tls available t schl administratrs reductins in class size appear amng the mst prmising prescriptins fr imprving student achievement. Hwever, the statistical evidence n the linkages between class size and student perfrmance is mixed. 1 Since the publicatin f the influential Cleman reprt (1966), there have been literally hundreds f studies examining the relatinship between class size and student achievement. The results span the full range f pssible cnclusins: sme find that there is a significant and psitive relatinship between class size and student achievement; sme find that smaller classes are mre effective; sme find that there is n discernible relatinship. Inevitably, sme f the uncertainty in the literature derives frm the fact that there is n unifrmly agreed specificatin f the mdel r estimatin methd fr the causal effect f class size. Mst empirical studies have emplyed least squares methds t btain estimates f the effect f class size n student achievement, and thus present a mean treatment view f class size effect. Recgnizing the hetergeneity in the ptential effects several authrs have recently suggested that a mre disaggregated estimatin f the plicy effects wuld be preferred, see e.g. Hanushek (1986), Krueger (1997), Card (2001) and Angrist and Krueger (2001). Hwever, t the best f ur knwledge, nly tw studies take up the challenge t investigate class size effects acrss quantiles f schl achievement distributin. 1 Fr meta-analysis, see Glass and Smith (1979), Glass et al. (1982), Prwll (1978), Rbinsn and Wittebls (1986) and Hanushek (1998). See als, Summers and Wlfe (1977), Hanushek (1986,1997), Angrist and Lavy (1999) and Krueger (2003). The Tennessee Student/Teacher Achievement Rati experiment, knwn as prject STAR, invlved 11,600 students frm 80 schls ver fur years Finn and Archilles (1990). Initiated in 1996, the Califrnia Class Size Reductin, namely the CSR prgram, cst ver $1 billin per year and affected ver 1.6 millin students (Class Size Reductin in Califrnia: Early Evaluatin Findings: , 1999). Dutch plicy makers have recently dedicated mre than $500 millin t reduce class sizes in primary educatin (Levin, 2001).

19 Lingjie Ma and Rger Kenker 19 Eide and Shwalter (1998) using US data, apply quantile regressin methds t a mdel f student achievement and find that the class size effect is insignificantly different frm zer at all quantiles f students achievement distributin. It shuld be emphasized that this mdel des nt include students family backgrund, r peer effects, and that they treat the class size variable as exgenus. Nting the endgeneity prblem, Levin (2001) applies a variant f Amemiya s (1982) methds t a structural equatin mdel, but als finds little empirical supprt fr beneficial effects f smaller classes at mst quantiles with r withut peer effects added t the mdel. Nte that bth Eide and Shwalter (1998) and Levin (2001) present what we have characterized as a mean quantile treatment effect view f class size effects: Hw des mean class size affect the distributin f academic utcmes? By revealing the variatins f class size effects acrss quantiles f students achievement, the MQTE apprach ffers a mre cmplete view than earlier wrk. Hwever, the effect f variatins acrss quantiles f the distributin f class sizes remains bscure. As a cnsequence, it is hard t evaluate the class size effect withut acknwledging that varius class sizes have different influences n students academic perfrmance. Fr brader view f class size effects, we cnsider the structural quantile treatment effect in the framewrk that we have set ut in Sectin 2, in an effrt t explre the ptential hetergeneity in the class size effect ver bth the distributin f students achievement as well as the distributin f class sizes Data Descriptin. The data we emply is the first wave f the PRIMA chrt study, which cntains detailed infrmatin n Dutch primary schl students in grades 2, 4, 6, and 8 as well as the assciated teacher and schl characteristics fr the schl year 1994/ The PRIMA chrt study is a cmprehensive survey f primary educatin in Hlland, enabling researchers t explre relatinships between pupil s achievements, their characteristics, thse f their parents, as well as class level and schl level characteristics. Pupils are tested with regard t intelligence, reading abilities, the Dutch language and mathematics. Backgrund data are gathered thrugh parents and teachers and detailed schl level data are furnished by the directrs f the participating schls. In ttal, there are abut 57,000 pupils frm 700 primary schls in the survey. Of these, 450 schls frm the representative randm sample that we use in this paper. Only grades 4, 6 and 8 are cnsidered and the three grades are pled tgether in ur analysis. 3 A brief statistical summary f the variables used in ur mdeling is reprted in Table 6.1. The average class size is 24 and ranges frm 5 t 39, but abut 70% f classes are between It may be nted that the variability f math scres is cnsiderably higher than that f the language scres. Abut 72% f the schls in the sample are public, but it prbably shuld be emphasized that the distinctin between private and public schls in Hlland is nt nearly s great as ne may be led t expect frm the vantage pint f the US. Estimates f the interactin f schl 2 This data has been previusly used by Dbblelsteen et al (1998) and Levin (2001). 3 The ages f pupils in grade 4, 6 and 8 are arund 7 8, 9 10 and 11 12, respectively.

20 20 Quantile Regressin Methds fr Structural Mdels Minimum Maximum Mean Std. Dev. Language Scre Math Scre Pupil s Gender (Female=1) IQ Sci-Ecnmic Status (SES) Risk Peer Effects (Language) Peer Effects (Math) Class Size Teacher s Experience (Years) Schl Denminatin (Public = 1) Weighted Schl Enrllment (WSE) Table 6.1. Sample Summary Statistics: There are 5698, 5368 and 5608 bservatins fr grade 4, 6, and 8, respectively, which after pling and deleting cases with missing values fr imprtant variables yielded 12,203 bservatins. denminatin and class size indicate that there is n significant difference in class size effects between public and private schls Mdel Specificatin. Befre cnsidering the frmal mdel, there are tw cncerns abut class size effects that shuld be addressed. The first ne is the causal mechanism: class size per se shuld nt cntribute t students academic achievement. Presumably, class size perates thrugh varius channels that exert influences n student perfrmance. Fr example, smaller classes may induce changes in instructinal methds and change the nature f peer effects. Bth these factrs are thught t play imprtant rles in students academic perfrmance. Lazear (2001), fr example, has fcused n the public gd aspect f classrm teaching and investigates the cngestin effects f class size frm a theretical perspective. But there seems t be n generally accepted thery f the causal mechanism that links class size t student perfrmance. A secnd majr cncern fr the emprical study f class size effects is ptential endgeneity. Parents may make lcatin decisins based n the quality f lcal public schls attempting t ensure that their children attend small classes; schl administratrs may have a desire t put the lwer attainment students in smaller classes r try t assign better teachers t bigger classes. Crrespndingly, t treat the endgeneity prblem f class size, there are tw appraches in the literature: ne is t sidestep endgeneity issues by fcusing n experimental settings like the Tennessee STAR experiment, r related natural experiments as in Hxby (2000); the ther is t use instrumental variable methds t crrect fr the bias induced by endgenus cvariates, e.g., Krueger (1997), Angrist and Lavy (1999), Hanushek (2001). While mst

21 Lingjie Ma and Rger Kenker 21 studies adpt the IV apprach, a gd IV is ntriusly hard t find. Empirically, researchers have taken the assigned class size, Krueger (1997); schl enrllment, Akerheilm (1995), Iacvu (2001), Levin (2001); and grade enrllment, Angrist and Lavy (1999); as instrumental variables fr actual class size in either cntinuus r nn-cntinuus frms. Given the bservatinal, i.e. nn-experimental, nature f ur data, we may begin by cnsidering a cnventinal apprach based n a linear structural equatin mdel f the frm (6.1) (6.2) y = α 0 + X i α 1 + X c α 2 + X s α 3 + Y δ + u Y = β 0 + X i β 1 + X c β 2 + X s β 3 + Zγ + U. The precise specificatin f the randm cmpnents u and U will be delayed mmentarily while we cnsider the bservable variables. Math r language test scres are dented by y i fr student i in class c and schl s; X i are individual i s characteristic variables including pupil s gender, IQ, sciecnmic status (SES), peer effects and risk level; 4 X c are class c s characteristic variables including teacher s experience; 5 X s are schl s s characteristic variables, including the schl denminatin (public r nnpublic) nly; Y is the cvariate fr class size and Z dentes the instrument fr class size; u and U dente unbserved randm cmpnents. As we have already nted, in the pure lcatin shift frm f the mdel the structural effect f class size is unambiguus: the parameter δ captures this effect and it may be interpreted as the shift in lcatin f test scres induced by a change in class size that describes the effect at all quantiles f the academic perfrmance distributin and at all quantiles f the class size distributin. What is z, the instrumental variable fr class size? The Dutch Ministry f Educatin impsed a new funding allcatin rule during the time perid f the first wave f the PRIMA survey. Each primary schl reprted weighted schl enrllment (WSE) t the Ministry with weights determined by the sci-ecnmic status f the enrlled students. Based n the value f this WSE, the Ministry allcated funding t each schl and this funding determined hw many teachers the schl culd hire. It is clear that this WSE variable is clsely related t the actual class size but has n direct relatin with student achievements cnditinal n characteristics. Fllwing Levin (2001) we emply WSE as ur instrumental variable fr class size. 4 Students are defined as at risk based n bserved cgnitive and/r behaviral prblems. Schl must dcument students prblems regularly. Based n infrmatin frm the student prfiles, each student is given a scaled scre ranging frm 1 t 5 in ascending rder f riskiness. Fr detailed infrmatin n sci-ecnmic status (SES), see Levin (2001), fr the simplicity, we take recde SES as binary, with 1 indicating higher SES. The peer effect is measured by the classmates average test scre. 5 Preliminary estimatin indicated that teachers age, sex and level f educatin were insignificant influences n students achievement.